On Ryser’s conjecture: t-intersecting and degree-bounded hypergraphs, covering by heterogeneous sets

A famous conjecture (usually called Ryser’s conjecture), appeared in the Ph.D thesis of his student, J. R. Henderson [9], states that for an r-uniform r-partite hypergraph H, the inequality τ(H) ≤ (r − 1)·ν(H) always holds. This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig’s theorem [16], and in the case of r = 3, which was proved by Aharoni in 2001 [2]. Here we study some special cases of Ryser’s conjecture. First of all the most studied special case is when H is intersecting. Even for this special case, not too much is known: this conjecture is proved only for r ≤ 5 in [8, 19]. For r > 5 it is also widely open. Generalizing the conjecture for intersecting hypergraphs, we conjecture the following. If an r-uniform r-partite hypergraph H is t-intersecting (i.e., every two hyperedges meet in at least t < r vertices), then τ(H) ≤ r − t. We prove this conjecture for the case t > r/4. Gyárfás [8] showed that Ryser’s conjecture for intersecting hypergraphs is equivalent to saying that the vertices of an r-edge-colored complete graph can be covered by r − 1 monochromatic components. ∗Research is supported by a grant (no. K 109240) from the National Development Agency of Hungary, based on a source from the Research and Technology Innovation Fund.